Abstract
A graph is total domination edge-critical if the addition of any edge decreases the total domination number, while a graph with minimum degree at least two is total domination vertex-critical if the removal of any vertex decreases the total domination number. A 3 t EC graph is a total domination edge-critical graph with total domination number 3 and a 3 t VC graph is a total domination vertex-critical graph with total domination number 3. A graph G is factor-critical if G − v has a perfect matching for every vertex v in G. In this paper, we show that every 3 t EC graph of even order has a perfect matching, while every 3 t EC graph of odd order with no cut-vertex is factor-critical. We also show that every 3 t VC graph of even order that is K 1,7-free has a perfect matching, while every 3 t VC graph of odd order that is K 1,6-free is factor-critical. We show that these results are tight in the sense that there exist 3 t VC graphs of even order with no perfect matching that are K 1,8-free and 3 t VC graphs of odd order that are K 1,7-free but not factor-critical.
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Research supported in part by the South African National Research Foundation and by a grant from the Harry Oppenheimer Trust.
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Henning, M.A., Yeo, A. Perfect Matchings in Total Domination Critical Graphs. Graphs and Combinatorics 27, 685–701 (2011). https://doi.org/10.1007/s00373-010-1000-3
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DOI: https://doi.org/10.1007/s00373-010-1000-3